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Continued Logarithms And Associated Continued Fractions

Borwein, Jonathan M. and Calkin, Neil J. and Lindstrom, Scott and Mattingly, A. (2015) Continued Logarithms And Associated Continued Fractions. (Submitted)

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    Abstract

    We investigate some of the connections between continued fractions and continued logarithms. We study the binary continued logarithms as introduced by Bill Gosper and explore two generalizations of the contin- ued logarithm to base b. We show convergence for them using equivalent forms of their corresponding continued fractions. Through numerical ex- perimentation we discover that, for one such formulation, the exponent terms have finite arithmetic means for almost all real numbers. This set of means, which we call the logarithmic Khintchine numbers, has a pleas- ing relationship with the geometric means of the corresponding continued fraction terms. While the classical Khintchine’s constant is believed not to be related to any naturally occurring number, we find surprisingly that the logarithmic Khintchine numbers are elementary.

    Item Type: Article
    Subjects: UNSPECIFIED
    Faculty: UNSPECIFIED
    Depositing User: Mrs Naghmana Tehseen
    Date Deposited: 28 Mar 2016 12:13
    Last Modified: 28 Mar 2016 12:13
    URI: https://docserver.carma.newcastle.edu.au/id/eprint/1687

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